JHEP09(2012)005

Published for SISSA by Springer

Received: April 4, 2012

Revised: June 11, 2012

Accepted: August 12, 2012

Published: September 4, 2012

An expansion for neutrino phenomenology

Benjamın Grinsteina and Michael Trottb

aDepartment of Physics, University of California,

San Diego, La Jolla, CA 92093 U.S.A.bTheory Division, Physics Department, CERN,

CH-1211 Geneva 23, Switzerland

E-mail: bgrinstein@ucsd.edu, michael.trott@cern.ch

Abstract: We develop a formalism for constructing the Pontecorvo-Maki-Nakagawa-

Sakata (PMNS) matrix and neutrino masses using an expansion that originates when a

sequence of heavy right handed neutrinos are integrated out, assuming a seesaw mecha-

nism for the origin of neutrino masses. The expansion establishes relationships between the

structure of the PMNS matrix and the mass differences of neutrinos, and allows symmetry

implications for measured deviations from tri-bimaximal form to be studied systematically.

Our approach does not depend on choosing the rotation between the weak and mass eigen-

states of the charged lepton fields to be diagonal. We comment on using this expansion

to examine the symmetry implications of the recent results from the Daya-Bay collab-

oration reporting the discovery of a non zero value for θ13, indicating a deviation from

tri-bimaximal form, with a significance of 5.2σ.

Keywords: Neutrino Physics, Solar and Atmospheric Neutrinos

ArXiv ePrint: 1203.4410

Open Access doi:10.1007/JHEP09(2012)005

JHEP09(2012)005

Contents

1 Introduction 1

2 Constructing a flavour space expansion for neutrino phenomenology 3

2.1 Review of seesaw the mechanism 3

2.2 Perturbing the seesaw 4

2.2.1 Model dependence of perturbing the seesaw 4

2.3 Developing the seesaw perturbations 6

2.4 Inverted and normal hierarchy and flavour space 8

2.4.1 The expansion without flavour alignment 9

2.4.2 The expansion with flavour alignment 9

2.5 UPMNS and flavour space 10

3 (Un)relating flavour symmetries to the UPMNS form 11

3.1 Perturbative breaking of T B form 12

4 Conclusions 13

1 Introduction

The standard theory of neutrino oscillations, where three mass eigenstate neutrinos differ

from their interaction eigenstates leading to the observed neutrino oscillations, is consistent

with current experimental data. The amplitude of the oscillations among various neutrino

species is related to the misalignment of the interaction and mass eigenstates of the neutri-

nos, characterized by the Pontecorvo-Maki-Nakagawa-Sakata (PMNS) matrix [1, 2]. Taking

the relation between the interaction (primed) and mass (unprimed) eigenstates to be given

by f ′L/R = U(f, L/R) fL/R, where f = {ν, e, µ, · · · }, the PMNS matrix is given by

UPMNS = U†(e, L)U(ν, L). (1.1)

The UPMNS matrix can be parameterized in terms of three angles θ12, θ13, θ23 and three

CP violating phases δ, α1,2. Defining sij = sin θij and cij = cos θij with the conventions

0 ≤ θij ≤ π/2, and 0 ≤ δ, α1,2 ≤ 2π a general parameterization of this matrix is given by

UPMNS =

1 0 0

0 c23 s230 −s23 c23

× c13 0 s13e

−iδ

0 1 0

−s13e−iδ 0 c13

× c12 s12 0

−s12 c12 0

0 0 1

P, (1.2)

where P = diag(eiα1/2, eiα2/2, 1) is a function of the Majorana phases, present if the right

handed neutrinos are Majorana, while δ is a Dirac phase. This latter phase can contribute in

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JHEP09(2012)005

principle to neutrino oscillation measurements, while the Majorana phases cannot. Recent

global fit results on neutrino mass differences and measured mixing angles using old/new

reactor fluxes are given in ref. [3], (with new reactor flux results in brackets):

∆m221 = (7.58+0.22

−0.26)× 10−5eV2,

|∆m232| = (2.35+0.12

−0.09)× 10−3eV2,

sin2(θ12) = 0.306+0.18−0.15 (0.312+0.17

−0.16),

sin2(θ13) = 0.021+0.007−0.008 (0.025+0.007

−0.007),

sin2(θ23) = 0.42+0.08−0.03. (1.3)

The error is the reported 1σ error. Note that ∆m232 ≡ m2

3−(m21+m2

2)/2 and ∆m232 > 0, (<

0) corresponds to a normal (inverted) mass spectrum. This pattern of experimental data

is perhaps suggestive of a PMNS matrix that has at least an approximate “tri-bimaximal”

(T B) form [4]. Fixing sin2(θ12) = 1/3 and θ13 = 0 the T B form is

UPMNS ≈ UTB =

√

23

1√3

0

− 1√6

1√3− 1√

2

− 1√6

1√3

1√2

, (1.4)

for a particular phase convention.

The structure of this matrix could be fixed by underlying symmetries. In attempting

to determine such an origin of this matrix, the “flavour” basis where one assumes U(e, L) =

diag(1, 1, 1) is frequently used. When this assumption is employed the relationship between

the weak and mass neutrino eigenstates is identified with the UPMNS , i.e. ν ′i = (UPMNS)ij νjfor i, j = 1, 2, 3. There have been many attempts to link the approximate T B form of the

neutrino mass matrix to symmetries of the right handed neutrino interactions in this basis,

see ref. [5, 6] for a review. Recent experimental results provide evidence for deviations from

this T B form. Evidence for sin2(θ13) 6= 0 in global fits is reported to be > 3σ in ref. [3] at

this time. As this paper was approaching completion, the discovery of non-vanishing θ13was announced by the Daya Bay Collaboration [7] with a reported value of

sin2(2 θ13) = 0.092± 0.016(stat)± 0.005(syst), (1.5)

corresponding to 5.2σ evidence for non zero θ13. It is reasonable to expect further specu-

lation about the origin of the deviation from T B form in light of this result, where again

the flavour basis will be frequently assumed.

There is no clear experimental support for assuming that U(e, L) = diag(1, 1, 1). This

choice can be motivated by an ansatz related to the origin of the approximately diagonal

structure of the Cabibbo-Kobayashi-Maskawa (CKM) matrix, and a further ansatz on

the relation between U†(d, L) and U†(e, L), see ref. [5] for a coherent discussion on this

approach. This choice can also be justified with model building in principle, see ref. [8] for

an example. Conversely, in grand unified models frequently associated with the high scale

involved in the seesaw mechanism, the quark and lepton mass matricies can be related in

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JHEP09(2012)005

such a manner that the flavour basis cannot be chosen, or at least, the choice of the flavour

basis can be highly artificial.

It is of interest to have a formalism for neutrino phenomenology that is as robust and

basis independent as possible. Clearly linking symmetries to the form of the PMNS matrix

requires that the physical consequences of a symmetry are not dependent on a basis that

can be arbitrarily choosen for U(e, L), such as the flavour basis. In this paper, we develop a

perturbative approach to the structure of the PMNS matrix as an alternative to symmetry

studies that are basis dependent.1 Using this approach we show a basis independent rela-

tionship between the eigenvectors that give U(ν, L) and U(e, L) corresponding to T B form

at leading order in our expansion. We also examine the prospects of relating patterns in

the data in low scale measurements of neutrino properties with high scale flavour symmetry

breaking, illustrating how our approach can be used to study the recent reported discovery

of non zero θ13 reported in ref. [7].

2 Constructing a flavour space expansion for neutrino phenomenology

In this section we develop a formalism linking the measured differences in the neutrino mass

eigenstates with the structure of the PMNS matrix, which we will refer to as a flavour space

expansion (FSE). There is an experimental ambiguity in the measured mass hierarchies at

this time. The neutrino mass spectrum can be a normal hierarchy (m3 > m2 & m1) or

an inverted hierarchy (m3 < m1 . m2). In our initial discussion we will assume a normal

hierarchy. The formalism can be reinterpreted for an inverted hierarchy.

2.1 Review of seesaw the mechanism

Recall the standard seesaw scenario [17–19], with three right handed neutrinos NRi.2 We

use the notation ` = (νL, eL) for the left handed SU(2) doublet field and eR for the SU(2)

singlet lepton field carrying hypercharge. These fields carry flavour indicies i, j.3 We also

define H = i τ2H?, where H is the Higgs doublet of the SM, with 〈HT 〉 = (0, v/

√2). Then

the lepton sector of the Lagrangian in the seesaw scenario is the following

L = i N ′Ri ∂/N′Ri −

1

2N ′cR iM

′ij N

′Rj − ¯′

LiH (y′E)ij e′Rj − N ′Ri (y′ν)ij H

† `′Lj + h.c. , (2.1)

here we have defined N cRi = CNT

Ri with C the charge conjugation matrix. There is freedom

to rotate to the mass basis by introducing the unitary rotation matricies U(f, L/R), so

that

N ′Ri = U(N,R)ij NRj , `′Li = U(L,L)ij `Lj , e′Ri = U(e,R)ij eRj . (2.2)

1This is a modern implementation in the neutrino sector of an old idea of relating the mixing matricies

of the standard model (SM) to the measured quark or neutrino masses, for pioneering studies with this aim

see ref. [9, 14–16].2Our initial discussion will largely follow ref. [20–22].3These flavour indices will run over a, b, c for the three N ′

i . We use this notation to distinguish these

flavours from the measured mass differences of the physical eigenstates (m1,2,3) and also the index on the

Yukawa vectors of each N ′i , which run over 1, 2, 3.

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JHEP09(2012)005

The kinetic terms are unchanged by these rotations, and one is free to further rotate to a

basis in the flavour space of M, yE , yν defined by the set of all possible M, yE , yν related

through

M ′ → U(N,R)?M U(N,R)†,

y′E → U(L,L) yE U(e,R)†, (2.3)

y′ν → U(N,R) yν U(L,L)†.

We choose to make the initial rotation to a basis where M is diagonal real and nonnegative.

This fixes U(N,R). Initially the Majorana mass matrix has three complex eigenvalues

Ma,b,c = ηa,b,c |Ma,b,c| = ηa,b,c (µa,b,c), here ηa,b,c are the Majorana phases. Working in this

basis [20–22] shifts any Majorana phases into the yν matrix and Ni =√ηiNRi +

√η?i N

cRi.

Integrating out the heavy Ni one obtains the dimension five operator [23]

L5 =1

2(H†`i)Cij (H† `j) + h.c. , (2.4)

with the matrix Wilson coefficient Cij = (yTν ηM−1 yν)ij . The key observation that we use

in this work is that when integrating out the Ni in sequence4 a perturbative expansion

for neutrino phenomenology that is related to a hierarchy in the magnitude of the contri-

butions to Cij can be constructed. Note that this is also the key point in the sequential

dominance idea of refs. [10–13], however, the formalism we will develop is distinct from

these past results.

2.2 Perturbing the seesaw

2.2.1 Model dependence of perturbing the seesaw

The FSE we develop depends on the mass spectrum and the Yukawa couplings of the Ni.

Also, in principle, the flavour orientation5 effects the quality of the FSE. Experimentally,

at this time, all of these Lagrangian parameters are individually unknown, thus we will be

forced to assume that a perturbative expansion of the form we employ exists. Our approach

is to view the seesaw Lagrangian given in eq. (6) as an effective theory, and we do not

attempt to uniquely identify and restrict ourselves to a particular UV physics that dictates

the low energy parameters in this effective theory in this paper. However, the formalism

we develop allows perturbative investigations of the neutrino mass spectrum and mixing

angles in many scenarios and is in fact quite general. It would be surprising if all of the

unknown parameters conspired to forbid an expansion of this form from being present. We

seek to illustrate this point in this section, by demonstrating a simple scenario where the

FSE would be of some utility.

4We distinguish in this paper right handed neutrinos that are in a mass diagonal basis, reserving the

notation Ni for such states, compared to NRi states, which are not mass diagonal in general.5By the flavour orientation of the Yukawa couplings, we mean the orientation of the Yukawa coupling

vectors ~x, ~y, ~z with respect to the leading order eigenvectors ~ρa,b,c; see the next section for definitions and

further details.

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JHEP09(2012)005

There are of course cases when the formalism we outline cannot be used. When the

µa,b,c and Yukawa couplings of the Ni are both highly degenerate a FSE cannot be used.6

This is also the case if a non-degenerate pattern of the µa,b,c and the Yukawa couplings

are such that the perturbations to the neutrino masses are similar in magnitude. However,

as the µa,b,c depend on the matching of the effective theory to UV physics, while the

Yukawa couplings are dimension four and are not UV sensitive, this would require tuning

the physics of different energy scales. Unless model building is used to relate the masses

and Yukawa couplings of the Ni so that the contributions to the νL masses are the same,

a FSE is expected quite generally, and can be related to neutrino mass differences, as we

will show.

One can study the expected spectrum of left handed masses due to eq. (2.1) in generic

scenarios. Without assuming other interactions beyond the SM, the N ′Ri are not distin-

guished by any quantum number. As a result, the non-diagonal mass matrix that is the

coefficient of the operator matrix Oij = N′RiN

′Rj given by M ′ij is naively expected to have

entries that are all similar in magnitude. As the operator is dimension three, the mass

matrix is expected to be proportional to the highest scale UV physics (violating lepton

number) that was integrated out leading to this effective theory. We denote this scale by

M0 and the naive non-diagonalized mass matrix in this case as

M ′ij 'M0

1 1 1

1 1 1

1 1 1

, (2.5)

with eigenvalues µ0i = {3M0, 0, 0}. The vanishing of the two eigenvalues of M ′ij can be

lifted by interactions of the N ′Ri. These interactions can be dictated by beyond the SM

quantum numbers assigned in UV model building. If the mass matrix is still approximately

degenerate, as in eq. (2.5), then a hierarchy of the Ni masses is still expected. An example

of a small breaking where other interactions do not need to be assumed is given by the

orientations of the N ′Ri in flavour space. Rotating to the lepton diagonal mass basis, these

interactions give loop corrections

εij = δM ′ij/M′ij '

(y′ν)∗ik(y′ν)jk

16π2log

(µ2

M20

)no sum on i, j. (2.6)

Logarithmic corrections of this form are required to cancel the renormalization scale depen-

dence of the pole masses of the N ′Ri. Including such corrections leads to M ′ij 'M0 (1+εij).

These corrections split the mass spectrum; diagonalizing one finds

µ1 = M0

(3 +

∑ij εij

3

), µ2 =

2

3M0

∑i

εii −∑i<j

εij

, µ3 = M0O(ε2). (2.7)

A hierarchical spectrum of µi is expected with the pattern (µ1, µ2, µ3) 'M0(1, ε, ε2).

6An exception to this statement is when the flavour orientation is such that an expansion is present due

to the geometry in flavour space, we discuss an example of this form in section 2.4.

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JHEP09(2012)005

In the case of degenerate µi due to the M ′ij mass matrix not conforming to these generic

expectations, the expansion can follow from a hierarchy in the Yukawa couplings of the Ni

— such as the hierarchical pattern of Yukawa couplings in the SM. Lastly, the suitability of

the expansion can follow only from the orientation in flavour space of the Yukawa coupling

vectors of the Ni. In summary, the appropriateness of the FSE is clearly model dependent,

but we expect it to be broadly applicable in realistic models.

2.3 Developing the seesaw perturbations

Consider supplementing the SM field content by a single right handed neutrino7 Na and an

interaction term that couples it into a linear combination of `βL. The coupling is fixed by the

complex Yukawa vector ~xT = {x1, x2, x3} in flavour space whose form can be constrained

by a flavour symmetry, but is here left arbitrary. We absorb the overall Majorana phase

into this vector. The relevant Lagrangian terms8 are given by

La = −µa2NaNa − xβ Na H

† `βL + h.c. (2.8)

The right handed neutrino can be integrated out, giving for the left handed neutrino mass

matrix a nonzero eigenvalue

M≡ v2

2µa~x~xT = U(ν, L)? diag(0, 0,ma)U(ν, L)†. (2.9)

The matrix U(ν, L) is the matrix (~ρc?, ~ρb

?, ~ρa?) of normalized (column) vectors ~ρ ?a,b,c that

solve M~ρ ? = m~ρ, with m real and non-negative.9 These vectors are also eigenvectors of

M†M = (~x?~x†) (~x~xT ) = (~x†~x)~x? ~xT with eigenvalues m2; their complex conjugates, ~ρ are

eigenvectors of MM† = (~x~xT ) (~x? ~x†) = (~x†~x)~x~x†, also with eigenvalues m2. One finds

the leading eigenvector and eigenvalue

~ρa = ~x/|~x|, ma = v2(~x†~x)/2µa. (2.10)

Here µa is real and non-negative due to the initial flavour basis choice that fixed U(N,R),

while U(ν, L) and ~x are in general complex.10 The remaining two eigenvectors ~ρb,c are

such that 〈 ~ρa|~ρb〉 ≡ ~ρa† ~ρb = 0, 〈 ~ρa|~ρc〉 = 0. These eigenvectors will lead to the remaining

(smaller) mass eigenvalues, and will perturb the leading eigenvalue and eigenvector and thus

the U(ν, L) matrix. This is the expansion we seek to exploit. Consider the perturbation

that generates the second eigenvalue of the neutrino mass matrix due to a second right

handed neutrino Nb, which we define to couple into a linear combination of the `L given

7The subscript in Ni will run over labels a, b, c; for now we are just considering the single spinor field

Na.8We have chosen U(N,R) as discussed in section 2.1 to rotate to a mass diagonal basis for Ni and

absorbed this rotation into a redefinition of the Yukawa matrices yν . The Yukawa vectors ~x, ~y and ~z

introduced below are given in this basis.9Here we use the notation that the eigenvectors run over the flavour indicies a, b, c to denote that they

are associated with integrating out each of the Na,b,c at leading order in the corresponding mass eigenvalues.10This choice is allowed by the unitary flavour transformations that are symmetries of the kinetic terms,

eqs. (6–8).

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JHEP09(2012)005

by ~yT = {y1, y2, y3}. One obtains a second eigenvalue in the left handed neutrino mass

matrix so long as ~y 6‖ ~x. The perturbation of MM† is given by

(M+ δM) (M+ δM)† =v4

4µ2a~x~xT (~x? ~x†) +

v4

4µa µb(~x~xT ) (~y? ~y†) +

v4

4µa µb(~y ~yT ) (~x? ~x†),

+v4

4µ2b~y ~yT (~y? ~y†). (2.11)

At leading order the ~ρb,c have degenerate (vanishing) eigenvalues. One is free to rotate to a

chosen basis in these vectors. We rotate to a basis in these vectors such that the following

perturbation vanishes for ~ρc

〈~ρc| (yy†) (yT y?) |~ρc〉 = 0. (2.12)

With this choice ~ρc retains a vanishing eigenvalue when Nb is integrated out. The eigen-

vector ~ρc should be orthogonal to ~ρa ∝ ~x. A normalized eigenvector basis at leading order

is then

~ρb =~x? × (~y × ~x)

|~x| |~x× ~y|, ~ρc =

~y? × ~x?

|~x× ~y|. (2.13)

Now we can determine the perturbation on the leading order eigenvectors and eigenvalues

when Nb is integrated out. The corrections to the eigenvalues and eigenvectors using

perturbation theory are given by

δ ~ρj = =∑i 6=j

〈~ρi|MδM† + δMM†|~ρj〉m2j −m2

i

~ρi,

δm2i = 〈~ρi|MδM† + δMM† + δM δM†|~ρi〉. (2.14)

Nonzero eigenvalues are obtained for mb,c at second order in the expansion due to the

orthogonal basis vectors causing the leading perturbations to each of these masses to vanish.

The leading perturbation to the eigenvectors and ma is first order in the expansion. We

retain the leading perturbation on the eigenvectors and the leading and subleading effects

on the masses to obtain nonzero eigenvalues. We find the following for the perturbations

δ ~ρa = µ2ab〈~ρb|~y〉(~y · ~x?)|~x|

δm2ab

~ρb, δm2a = 2µ2ab Re [〈~y|~x〉 (~x · ~y?)] + µ2bb |~y|4 cos2 θxy,

δ ~ρb = µ2ab〈~y|~ρb〉(~x · ~y?)|~x|

δm2ba

~ρa, δm2b = µ2bb|〈~ρb|~y〉|2 |~y|2,

δ ~ρc = 0, δm2c = 0. (2.15)

Here δm2ij = m2

i −m2j , µ

2ij = v4/(4µi µj) and cos θxy = |~x∗ · ~y|/|~x||~y| is a measure of (the

cosine of) the angle between the ~x, ~y Yukawa vectors. The masses are evaluated to the

appropriate order in the perturbative expansion and the eigenvector perturbations are in

general complex. Note that for the phenomenology of the UPMNS matrix that we will

study it will be sufficient to only retain the leading perturbation, while when studying the

mass spectrum the leading and sub-leading perturbations should be retained.

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JHEP09(2012)005

Finally integrate out the third right handed neutrino Nc with Majorana mass µc, which

couples into a linear combination of the `βL dictated by zTβ = {z1, z2, z3}. The resulting

eigenvector perturbations are

δ2 ~ρa = µ2ac〈~ρb|~z〉(z · x?)|~x|

δm2ab

~ρb + µ2ac〈~ρc|~z〉(z · x?)|~x|

δm2ac

~ρc,

δ2 ~ρb = µ2ac〈~z|~ρb〉(x · z?)|~x|

δm2ba

~ρa,

δ2 ~ρc = µ2ac〈~z|~ρc〉(x · z?)|~x|

δm2ca

~ρa. (2.16)

The mass perturbations are

δ2m2a = 2µ2ac Re [〈~z|~x〉 (~x · ~z?)] + µ2cc |~z|4 cos2 θxz,

δ2m2b = µ2cc |〈~ρb|~z〉|2 |~z|2,

δ2m2c = µ2cc |〈~ρc|~z〉|2 |~z|2. (2.17)

The measured masses of the neutrino’s are related to these perturbations as

m2A = m2

a + δm2a + δ2m

2a,

m2B = δm2

b + δ2m2b ,

m2C = δ2m

2c . (2.18)

It is interesting to note that a normal hierarchy emerges quite naturally from the FSE as

the leading neutrino mass ma receives corrections at linear order to its mass, while the

remaining masses only receive corrections at second order in the perturbations.

It is also important to note that this formalism does not require a hierarchy of the form

δ2m2i � δm2

i , only δ2m2i , δm

2i � m2

a is required. Expanding on this important point in

more detail, it is not required that the perturbation due to integrating out Nb is larger than

the perturbation due to integrating out Nc. Only that the effect of integrating out each

of these right handed neutrinos perturbs the initial mass matrix — which is dominated by

integrating out the initial right handed neutrino Na. The existence of these perturbations

are not necessarily direct statements on the relative size of the µi as we discuss in more

detail in the next section.

2.4 Inverted and normal hierarchy and flavour space

For a normal hierarchy (with notation m3 > m2 & m1) we identify (1, 2, 3) = (C,B,A) in

the equations above. The difference between the normal and inverted (m3 < m1 . m2) case

appears in the relative size of the δmi and δ2mi and the identification (C,B,A) = (3, 1, 2)

for an inverted hierarchy. An inverted hierarchy requires non generic perturbations in the

FSE, or one can trivially modify the expansion to only perturb when Nc is integrated out.

The size of δmi and δ2mi depends on the hierarchy in µa,b,c, the magnitude of the Yukawa

vectors, and also the orientation in flavour space of the vectors ~x, ~y, ~z. In this section, we

will discuss the size of the perturbations of the FSE in light of neutrino mass data in a

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JHEP09(2012)005

scenario where the perturbations are dictated primarily by a hierarchy in µa,b,c. We will

then discuss the case where the expansion arises primarily due to the orientation of the

Yukawa coupling vectors in flavour space.

2.4.1 The expansion without flavour alignment

We can examine the quality of the expansion by comparing to the measured mass differences

in neutrinos. Consider the case that the size of the perturbations is generic in the sense

that the Yukawa coupling vectors are taken to be O(1) with the orientation in flavour space

not significantly effecting the quality of the expansion. In this case, the expansion follows

from the relative size of the µi, i.e. µa < µb < µc. Consider the generic case in a normal

hierarchy. Using the FSE and retaining the dominant term

∆m232 = m2

a + δm2a + δ2m

2a − (δm2

b + δ2m2b) ≈ m2

a, (2.19)

while in the same manner δm2b ∼ ∆m2

21 and the expansion requires δ2m2c < ∆m2

32,∆m221.

Due to this, the expansion requires v2|~z|2/(2µc) <√

∆m232 ∼ 0.05 eV. This condition is

consistent with current bounds on the absolute neutrino mass scale, with a 95% C.L. bound

of∑mν = 0.28 eV quoted in ref. [24], assuming ΛCDM cosmology. It is also consistent

with current bounds from Tritium β decay experiments [25] which quote m(νe) < 2 eV at

95% C.L.

Expressing this condition in terms of the high mass scale of the Nc integrated out,

µc/|~z|2 & 1014 GeV. Generically one expects the mass scale of the right handed neutrino

operator to be the largest scale integrated out that violated L number, and this condition

for the lightest neutrino is clearly consistent with naive expectations of M0 ∼Mpl.

2.4.2 The expansion with flavour alignment

Now consider the case where the perturbative expansion follows from the flavour orientation

of the ~x, ~y, ~z vectors primarily. An example where this is the case is when the threshold

matching onto the UV physics is such that the Wilson coefficient matrix of Oij yields a

mass matrix with nearly degenerate eigenvalues. This occurs for example when

M ′ij 'M0

1 + ε ε ε

ε 1 + ε ε

ε ε 1 + ε

. (2.20)

In this case, a nearly degenerate mass spectrum of the NR follows, µa ' µb ' µc. Gener-

ating a normal or inverted hierarchy if one also has |~x| ∼ |~y| ∼ |~z| requires more precise

alignments in flavour space and allows a geometric interpretation of the measured neutrino

mass spectrum. In the FSE, the tree level masses of the SM neutrinos are

m2A = µ2aa|~x|4+2µ2ab|~x|2|~y|2 cos2 θxy+2µ2ac|~x|2|~z|2 cos2 θxz+µ2bb|~y|4 cos2 θxy+µ2cc|~z|4 cos2 θxz,

m2B = µ2bb|~y|4 cos2 θρby + µ2cc|~z|4 cos2 θρbz,

m2C = µ2cc|~z|4 cos2 θρcz. (2.21)

Consider the case that all of the µ2ij are similar in magnitude and |~x| & |~y| ∼ |~z| so that the

mass spectrum is primarily dictated by the orientation of the Yukawa vectors in flavour

space. An example of an inverted or normal hierarchy is shown in figure 1 in this case.

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JHEP09(2012)005

~x

~z

~y

~ρa

~ρb

−~ρc

~x

~z~y

~ρa

~ρb

−~ρc

Figure 1. A geometric interpretation of an inverted or normal hierarchy when µa ' µb ' µc and

|~x| & |~y| ∼ |~z|. Normal hierarchy on the left, inverted on the right.

2.5 UPMNS and flavour space

Now, assuming that a FSE expansion exists, let us consider its utility in examining the form

of UPMNS . The rotation matrix U(ν, L) with ~v?i = (~ρi + δ~ρi + δ2~ρi)? is given by U(ν, L) =

(~v?3, ~v?2, ~v

?1). The charged lepton mass matrix after electroweak symmetry breaking is given

by Me = v yE/√

2 and diagonalized by11

U(e,R)Me U†(e, L) = diag(me,mµ,mτ ). (2.22)

We take U(e, L)† =(~σ3†, ~σ2

†, ~σ1†)

, with ~σi the orthonormal column eigenvectors diagonal-

izing MeM†e. The expanded U?PMNS is of the form

U?PMNS =

~v3 · ~σ3 ~v2 · ~σ3 ~v1 · ~σ3~v3 · ~σ2 ~v2 · ~σ2 ~v1 · ~σ2~v3 · ~σ1 ~v2 · ~σ1 ~v1 · ~σ1

,

=

~ρ3 · ~σ3 ~ρ2 · ~σ3 ~ρ1 · ~σ3~ρ3 · ~σ2 ~ρ2 · ~σ2 ~ρ1 · ~σ2~ρ3 · ~σ1 ~ρ2 · ~σ1 ~ρ1 · ~σ1

+O(δ~ρi, δ2~ρi). (2.23)

This result makes clear that the first two right handed neutrinos that were integrated out in

the FSE contribute to the leading order structure of the PMNS matrix. The leading order

of the ~ρi eigenvectors only depended on ~x, ~y. As we have discussed, these neutrinos can be

integrated out in sequence, and for Yukawa couplings that are degenerate the lightest two

Ni can lead to the largest mass eigenvalues of the νL. In a mass degenerate case, they are

more strongly coupled to the `Lj . The unitarity of the UPMNS matrix is ensured when an

11The hierarchy of the charged lepton masses can be used to organize an expansion of U(e, L) in the

same manner by constructing M†eMe in principle. Conversely, in principle, one could employ an ansatz

that the hierarchy in these mass eigenvalues could be related to an expansion of U(e,R). As such we do

not employ a FSE on U(e, L)†. This ambiguity also limits the application of a FSE to the CKM matrix.

In this manner, the expansion we employ is most useful for expanding U(ν, L) when a seesaw mechanism is

the origin of the smallness of the neutrino masses.

– 10 –

JHEP09(2012)005

expansion of this form is employed by normalizing the ~vi eigenvectors order by order in the

expansion. Conversely, if the FSE is used when the expansion parameters are not small,

the constructed UPMNS matrix is not necessarily unitary.

It is important to note that, in general, it is the relationship between the eigenvectors

that determines UPMNS . Exact T B form at leading order in the FSE has a simple inter-

pretation. It follows from the relationship between the eigenvector ~ρ1 and the σi being

~ρ1 = (~σ1 − ~σ2)/√

2 in our chosen phase convention.

3 (Un)relating flavour symmetries to the UPMNS form

This formalism can be used to systematically explore symmetries that are consistent with

TB form. Consider an exact (T B) form of UPMNS at leading order in the FSE. This is

dictated by θ13 ≡ 0 and θ23 maximal. We restrict the Yukawa vectors in this discussion

to real values for simplicity. Including an unfixed s12 this gives (in a particular phase

convention)

UTB =

c12 s12 0

− s12√2c12√2− 1√

2

− s12√2c12√2

1√2

. (3.1)

Assuming UPMNS = UTB

~ρ1 =1√2

(~σ1 − ~σ2) , ~σ1 =1√2

(~ρ1 + c12 ~ρ2 − s12 ~ρ3),

~ρ2 =c12√

2(~σ1 + ~σ2) + s12 ~σ3, ~σ2 =

1√2

(−~ρ1 + c12 ~ρ2 − s12 ~ρ3),

~ρ3 = −s12√2

(~σ1 + ~σ2) + c12 ~σ3, ~σ3 = s12 ~ρ2 + c12 ~ρ3. (3.2)

Consider the “flavour” basis as an illustrative example of the FSE, where ~σ1 =

(0, 0, 1), ~σ2 = (0, 1, 0), ~σ3 = (1, 0, 0). In this case, U(e, L) = diag(1, 1, 1) and UPMNS =

U(ν, L). The flavour basis is appealing in that it allows the solution for the perturbations

to the ~ρi eigenvectors and the UPMNS matrix to proceed simply through solving

v2

2µa~x~xT +

v2

2µb~y~yT = U?TBdiag(0, δmb,ma+δma)U†TB, (3.3)

~x~xT +µaµb~y~yT =

(ma+δma)µav2

0 0 0

0 1 −1

0 −1 1

+

√2 δmb µav2

√

2 s212 s12 c12 s12 c12

s12 c12c212√2

c212√2

s12 c12c212√2

c212√2

.

Trivially one finds

~xT = (0,−1, 1)√

(ma + δma)µa/v, ~yT =(√

2 s12, c12, c12

)√δmb µb/v. (3.4)

It follows that in the flavour basis ~x · ~y = 0 so δma = 0 and (δ ~ρa, δ ~ρb, δ ~ρc) = (0, 0, 0). Now

consider including a third neutrino eigenvalue, retaining the required perturbation. For a

– 11 –

JHEP09(2012)005

solution, z2 = z3 is required, and consequently δ2m2a = 0 as ~x · ~z = 0. As a result the

leading eigenvector ~ρa is unperturbed by integrating out both Nb, Nc in the flavour basis,

one finds

~zT = ±√

2µc δ2mc

v

(√c212 +

δ2mb

δ2mcs212,

1√2

√δ2mb

δ2mcc212 + s212,

1√2

√δ2mb

δ2mcc212 + s212

).(3.5)

A µ ↔ τ symmetry implemented on ~y and ~z is consistent with T B form of the PMNS

matrix, as expected.

3.1 Perturbative breaking of T B form

Now consider the breaking of T B form. The value of θ13 measured by the DAYA-Bay

collaboration is in agreement with the global fit values given in ref. [3], as such, we use

the fit results to find the measured pattern of deviations in T B form. It is instructive to

construct the following ratios of experimental values characterizing the deviations of T Bform in each mixing angle. Using the small angle approximation

tan2(δθ12)

sin2(δθ13)= 0.02± 0.32

tan2(δθ23)

sin2(δθ13)= 0.01± 0.10

tan2(δθ12)

tan2(δθ23)= 2.0± 36. (3.6)

Here we have used the one sigma new reactor flux values of ref. [3] and taken a ± symmetric

one sigma error. Various breaking of T B form can be studied using the FSE and compared

to these results. Consider the case that Nb retains a µ ↔ τ flavour symmetry in its

couplings to the charged leptons, but Nc breaks such a symmetry so that deviations in T Bform are expected. Fix ~zT

′= ~zT + (0,∆1,∆2) with ∆1 6= ∆2 and treat this breaking as

a perturbation using the FSE. We use ∆m2AB ' ∆m2

AC assuming a normal hierarchy. At

leading order in the FSE the breaking of TB has the pattern

tan2(δθ12)

sin2(δθ13)= 0,

tan2(δθ23)

sin2(δθ13)=

2 cos2 θρbz + cos2 θρczcos2 θρbz + 2 cos2 θρcz

,

tan2(δθ12)

tan2(δθ23)= 0. (3.7)

As the range of the predicted value of tan2(δθ23)/ sin2(δθ13) is given by [0.5, 2] at leading

order in the FSE, this pattern of flavour breaking does not reproduce the data. In this

way, particular mechanisms of the breaking of T B form can be falsified. One can perform

the exercise of assuming T B form is broken in the flavour basis by Nb so that ~yT′

=

~yT + (0,∆1,∆2). Using the expansion one finds the pattern of deviations are distinct.

These breakings of T B form are also correlated with mass perturbations in each case

which are trivial to determine using this formalism.

We emphasize however that the relationships between the eigenvectors determine the

form of the PMNS matrix in general. This is easy to demonstrate in more detail. Consider

retaining a µ ↔ τ symmetry imposed on ~y and ~z but deviating from the flavour basis,

– 12 –

JHEP09(2012)005

choosing ~x, ~y, ~z as above but unfixing ~σi. At leading order in the expansion of UPMNS one

can solve for the condition that T B form is still obtained for general ~σi. One finds that

only the flavour basis for ~σi gives a valid solution at leading order in the expansion. This

makes clear that µ ↔ τ symmetry imposed on the Lagrangian alone is not related to T Bform in a basis independent manner.

As a further example that µ ↔ τ symmetry is also not unique or of particular physi-

cal significance in allowing T B form (with an appropriate choice on the σi eigenvectors),

consider the following procedure. Choose a (e, µ, τ) symmetry on the first interaction eigen-

vector, ie ~xT = (1, 1, 1)/√

3, and ~yT = (1, 1, 0)/√

2 as a simple interaction eigenvector for

Nb leading to an orthonormal eigenbasis at leading order, finding

U(ν, L) =

1√2

1√6

1√3

− 1√2

1√6

1√3

0 −√

23

1√3

. (3.8)

Solving directly for U(ν, L) so that at leading order T B form is obtained, one finds

U(e, L) =

16

(√2 + 2

√3)

16

(√2−√

3−√

6)

16

(√2−√

3−√

6)

16

(√2− 2

√3)

16

(√2 +√

3−√

6)

16

(√2 +√

3 +√

6)

−√23 −2+

√3

3√2

−√23 + 1√

6

. (3.9)

This procedure can be used for any flavour symmetry chosen to fix ~x, ~y in the FSE for theNi.

We also note that the impact of sterile neutrinos weakly coupled to the SM on neutrino

phenomenology can be systematically studied with this approach. For example, one can

relate any measured value of a deviation from T B form to the particular Yukawa coupling

vector of a single sterile neutrino, which can be shown to accommodate the value of θ13reported by the Daya-Bay collaboration while the three right handed neutrinos partners of

the SM fields give an exact T B form of the UPMNS matrix.

4 Conclusions

Flavour symmetries that are related to the structure of the UPMNS matrix only in a par-

ticular basis choice of U(e, L) can lead to suspect physical conclusions. As an alternative to

basis dependent symmetry studies, we have developed a perturbative expansion relating the

measured masses of the neutrinos to the form of the PMNS matrix. This expansion offers

a promising framework for broadly understanding the implications of the systematically

improving experimental neutrino data, particularly in a normal hierarchy.

We have illustrated our approach in an example where the flavour basis was chosen,

for the sake of familiarity, and then shown how the expansion can control the predictions

of T B form being broken. However, the approach we outline can accommodate any basis

choice. Indeed, it is the relationships between the eigenvectors that dictate the form of the

UPMNS matrix in a U(e, L) basis independent manner. This formalism can be employed in

model building to attempt to determine a compelling origin of the eigenvector relationship

that corresponds to T B form at leading order in the FSE. Further, as the breaking of T B

– 13 –

JHEP09(2012)005

form is now experimentally established due to the discovery of a non zero θ13 in ref. [7],

we expect the FSE to be of some phenomenological utility in falsifying mechanisms of the

breaking of the T B form of the UPMNS matrix, as the pattern of this breaking is further

resolved experimentally in the years ahead.

Acknowledgments

We thank Mark Wise for collaboration in the initial stages of this work. We also thank

Enrique Martinez for useful conversations and C. Grojean for comments on the manuscript.

The work of B.G. was supported in part by the US Department of Energy under contract

DOE-FG03-97ER40546. We thank the Aspen Centre for Theoretical Physics for hospitality.

Open Access. This article is distributed under the terms of the Creative Commons

Attribution License which permits any use, distribution and reproduction in any medium,

provided the original author(s) and source are credited.

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